M. Dellnitz, M. Golubitsky, A. Hohmann and I. Stewart
Spirals in Scalar Reaction Diffusion Equations
Intern. J. Bifur. & Chaos 5(6) (1995) 1487-1501
Spiral patterns have been observed experimentally, numerically
and theoretically in a variety of systems. It is often believed
that these spiral wave patterns can occur only in systems of
reaction-diffusion equations. We show, both theoretically
(using Hopf bifurcation techniques) and numerically (using both
direct simulation and continuation of rotating waves) that spiral
wave patterns can appear in a single reaction-diffusion equation
(in u(x,t)) on a disk, if one assumes `spiral' boundary conditions
(u_r=mu_theta). Spiral boundary conditions are motivated by assuming
that a solution is infinitesimally an Archemedian spiral near the
boundary. It follows from a bifurcation analysis that for this
form of spirals there are no singularities in the spiral pattern
(technically there is no spiral tip) and that at bifurcation
there is a steep gradient between the `red' and `blue' arms of
the spiral.